 Solution

Where $y\ne -1$, $y^{\prime }=-\sqrt{|1+y|}<0$. By the mean value theorem, the solution is strictly decreasing on every interval where it avoids $-1$. At $y=-1$, the derivative is zero, so a constant segment there is possible. Thus all solutions are nonincreasing, while strict decrease need not persist through an interval spent at that equilibrium.